Re: What is the 1st order formal system known as PA?


Nam Nguyen wrote:
> David C. Ullrich wrote:
>
> >
> >
> > The parts of this that make sense, and are not simple
> > misunderstandings on your part, have not been neglected,
> > they've very well known. To be more specific:
> >
> > You evidently think that the function symbols in FOL
> > actually refer to specific n-ary relations. They do not.
> >
> > They do not _until_ we start talking about an _interpretation_
> > of the language (a "model"). If we're talking about
> > interpretations of first-order PA then yes, "S" can have
> > many different interpretations. Nobody has every said
> > otherwise - it's awesomely obvious that "S" can have
> > many different interpretations.
> >
>
> Yes, it's "awesomely" obvious that S can have many different [model-]
> interpretations. But it seems much less obvious that, up-and-down
> the ladder of mathematical _introspection_, model-interpretation
> is *not* the only kind of legitimate interpretation, albeit the fact
> it's the only kind we *normally* have to deal with. For example,
> what about the semantic-interpretation of the language (or of a
> portion of it)? Are you sure any 2 reasoning beings can _always_
> agree what, say, a particular symbol of the language _mean_?
> And in this case, how can any 2 beings [human or not] guarantee
> that they semantically mean the same existence of a successor
> function? [And while for most of the familiar formulae such as
> Axy (x+y = y+x), that they might semantically mean 2 different
> successor functions - by the single name "S" - won't matter at all,
> how can we be so sure it won't matter to _all_ formulae of L(PA)?
> For instance, how can we be so sure it won't matter to GC(PA)?]
>

What do you mean by it "mattering"?

> >
> >>---Nam
> >
> >
> >
> > ************************
> >
> > David C. Ullrich
>
> --
> --------------------------------------------------------------------
> The difference between landscape and landscape is small,
> but there is a great difference between the beholders.
> Ralph Waldo Emerson
> ---------------------------------------------------------------------

.

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